Emergence of spatial structure (stripes, spots, hexagons, spirals) in spatially extended non-equilibrium systems via linear instabilities of a homogeneous base state. Core paradigms: Turing 1952 reaction-diffusion (activator-inhibitor…
pattern-formation
Swift-Hohenberg 1977 normal form for pattern onset
Swift-Hohenberg 1977 (Phys. Rev. A 15:319) proposed a minimal normal-form PDE capturing the essentials of Rayleigh-Bénard convection onset…
Real Ginzburg-Landau amplitude equation near pattern onset
Real Ginzburg-Landau equation derived by Newell-Whitehead-Segel 1969 for pattern onset in spatially-extended non-equilibrium systems. …
Fisher-KPP 1937 travelling wave with selected minimum speed
Fisher 1937 (Ann. Eugenics 7:355) and Kolmogorov-Petrovskii-Piskunov 1937 (Bull. Univ. Moscou A 1:1) independently introduced the 1D…
SH dispersion σ(k) = ε - (k² - k_0²)²: concave max at k_0, ε_c = 0
Linear stability analysis of the Swift-Hohenberg equation around the homogeneous base state ψ = 0. Linearised growth rate: σ(k) = ε - (k²…
Real GL amplitude: saturation |A|² = ε/g with residual ≡ 0
Steady-state of the real Ginzburg-Landau amplitude equation. Setting Ȧ = εA - gA³ = 0 gives two branches: A = 0 (trivial) and A = ±√(ε/g)…
Fisher-KPP c* = 2√(rD): derived from linearised dispersion discriminant
Derivation of the classical minimum wave speed. Linearise u_t = ru(1-u) + D·u_xx at u = 0 to obtain u_t = ru + D·u_xx; seek…